Exponential Functions
A free Algebra II lesson from the “Exponential Models” unit, with a worked example and practice problems including step-by-step solutions.
An exponential function has a variable in the exponent, such as y = a(b)^x. The base b is the multiplier from one step to the next. If b is greater than 1, the function shows growth; if b is between 0 and 1, it shows decay. Exponential functions matter because they model repeated percent change, compound interest, population growth, and depreciation. When practicing, identify the starting value and the multiplier before calculating. A common mistake is adding the same amount each step; exponential patterns multiply by the same factor instead.
What you'll learn
- Evaluate exponential functions
- Distinguish growth and decay
- Use percent change models
Worked example
Problem. Evaluate f(x) = 3(2^x) when x = 4.
- Substitute 4 for x.
- 2^4 = 16.
- 3 x 16 = 48.
Answer: 48
Practice problems
1. Evaluate f(x) = 2^x when x = 5.
Show solution
- Warm-up: First identify exactly what the question is asking: Evaluate f(x) = 2^x when x = 5.
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- 2^5 = 32.
- Check the result by substituting or estimating: the response should match 32 and make sense in the original problem.
Answer: 32
2. Evaluate g(x) = 5(2^x) when x = 3.
Show solution
- Warm-up: First identify exactly what the question is asking: Evaluate g(x) = 5(2^x) when x = 3.
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- 2^3 = 8.
- 5 x 8 = 40.
- Check the result by substituting or estimating: the response should match 40 and make sense in the original problem.
Answer: 40
3. y = 7(1.2)^x represents...
Choices: Growth · Decay · Linear change · No change
Show solution
- Core Practice: First identify exactly what the question is asking: y = 7(1.2)^x represents...
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- The base 1.2 is greater than 1.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Growth
4. y = 100(0.8)^x represents...
Choices: Decay · Growth · Quadratic change · No function
Show solution
- Core Practice: First identify exactly what the question is asking: y = 100(0.8)^x represents...
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- The base is between 0 and 1.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Decay
5. A value starts at 200 and grows by 10%. What multiplier is used each period?
Show solution
- Challenge: First identify exactly what the question is asking: A value starts at 200 and grows by 10%. What multiplier is used each period?
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Growth multiplier is 1 + 0.10.
- Check the result by substituting or estimating: the response should match 1.1 and make sense in the original problem.
Answer: 1.1
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